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I'm a undergrad(third year, Manchester uni and want to do a PhD) that is thinking of doing a PhD in this area or category theory in general.(Sorry for asking it here, Maths exchange stack didn't help as asked twice last week and then today, I spend like an hour a day worrying about what I'm studying and what I should study). Imagine a chess board and you need to move a piece, how you know you are moving the best piece?, now times that by a thousand.

Just wondering, what branches of Maths should I focus on? As I've been told that topology particular Algebraic topology is the main area for this. A lecturer told me I should focus on getting down Algebraic topology before thinking of doing category theory as most of the examples of category theory are from algebraic topology.

However, another lecturer told me I should be doing logic and particular model theory. I can take a course in it this year, but would mean not doing commutative algebra.

So before you delete this can you please tell me what I should be studying? As it would save me countless hours of worrying if I'm doing the right subjects.

P.S. I can do most of the stuff in Conceptual Mathematics category book. I don't understand topoi through.

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So you've asked two professors, in-person, and gotten two different answers. And now you've come to ask thousands of virtual mathematicians... –  Steve D Sep 18 '11 at 20:40
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I don't think mathematics works like that. There's no way you're going to get anywhere by learning only the bare minimum necessary for your particular field, if such a thing even exists. This is why you're getting such different answers. You're going to have to learn about all of those things to some extent, but probably you will only be able to learn one or two of them in-depth. (There's no point in deciding which ones now.) –  Dan Petersen Sep 18 '11 at 20:53
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@simplicity - what do you want to do with this hypothetical PhD of yours? Research or otherwise? If you are looking at higher category theory, I would guess the former, and in that case, one needs to at least absorb the 'vibe' of as much maths as you can. If you don't really love maths, then you may underestimate how much a PhD demands of you. –  David Roberts Sep 18 '11 at 22:19
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Sometimes I wonder if higher category theory is really the theme of so much of this new Century mathematics or if category theorists have just done a wide advertisement of their field on the internet... (or perhaps both...) –  Qfwfq Sep 18 '11 at 22:28
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Please do not cross-post: math.stackexchange.com/questions/65559/… –  Qiaochu Yuan Sep 18 '11 at 22:38
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closed as not a real question by Dan Petersen, Noah Snyder, Ryan Budney, quid, J.C. Ottem Sep 19 '11 at 8:21

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2 Answers

It seems to me that category theory (whether higher or not) may be the worst part of mathematics to approach with a narrow viewpoint --- just topology, or just algebraic geometry, or just logic, or just any one area. To me, much of the value and beauty of category theory lies in how it exhibits connections and similarities among many areas of mathematics.

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As many other have just said you cannot think to study just some particular subjects ignoring some other areas, expecially if you want to do research. Most of math was born from the observation of some similar phenomena in many different areas: for instance the concept of category itself was born from the observation that in math we deal every time with collections of structures and morphisms preserving those structures, that led to the abstraction of category, similarly I strongly doubt that Grothendieck could invent the concept of (generalized) sheaf if first he hadn't known the many concrete sheaves that appear in topology, differential geometry and algebraic geometry, so it couldn't get to the concept of (Grothendieck's) topos, and without that I'm not so sure that Lawvere could get to the concept of elementary topos while doing his research in logic. This are just some example of as math have evolved thanks to interaction of different areas (for instance, as you can see in the example above, from interaction of geometry and logic).

Just to answer to your comment about analysis there's a professor in Italy who studies higher dimensional category theory for his research in analysis, so analysis need higher category theory.

Of course the best place where you can get a lot of intuition of higher category theory is algebraic topology where higher categories are used to model homotopy types for topological spaces, via $\infty$-groupoids, and directed space, via $(n,r)$-categories where $n,r \in \omega \cup \{\infty\}$ but you can find a lot of higher dimensional category theory in logic and computer science too, I've seen some application in calculability theory and model theory where (higher) category theory is used to model the semantic of theories, in particular type theory (if you're interested in application of higher categorical logic-model theory you can take a look to Makkai's work and also Mike Shulman's work on homotopy type theory). Also in mathematical physics there are a lot of higher category theory as John Baez's work prove.

I suppose above you were referring to Cheng-Lauda "Illustrated guide book", that's a good book if you want to learn many approaches to $n$-categories, but in higher category theory there's a lot of more then just $(n,r)$-categories (like usually Mr.Shulman says), Leinster's "Higher operads, Higher categories" is more complete from this point of view because it presents a lot of stuff like generalized multicategories/operads or $fc$-multicategories. Anyway if you want some references on higher category theory you can find some here.

Hope this may help you.

(Edit: I've improved a little the answer now that I've found some other references.)

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simplicity999, the idea that an algebraic doesn't need to know analysis is wrong. First of all, in graduate school you likely will need to pass some qualifying exams that involve algebra, analysis and geometry. Second of all, major areas of algebra (e.g.,representation theory, number theory) get some inspiration from ideas in analysis. Other areas of algebra might not, but Blass is right in his answer that it's ironic someone with the intention of working in category theory would anticipate having a limited range of mathematical interests. –  KConrad Sep 19 '11 at 0:25
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@KConrad - assuming the student is in the US. For better or worse, not all countries have qual exams. –  David Roberts Sep 19 '11 at 1:00
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I think you're confusing the fact that most mathematicians only do research in one area, with the idea that mathematicians don't know the basics of other subjects. When you're talking about math at the upper undergrad level or first year graduate student level, that's stuff everyone is supposed to know (though, of course, many of us have forgotten parts of it). Specialization begins in earnest after a year or two of graduate school. –  Noah Snyder Sep 19 '11 at 2:05
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Dear Simplicity999, You impression about specialization may partly reflect your experience in the UK, where the time to complete a PhD is less than in the U.S. or (I believe) other European countries, so that students are forced to specialize early. But math is a highly interconnected subject, and it is not just the famous prize-winners who have a broad point of view. In the UK, my impression is that it is often the immediate post-PhD years in which people try to broaden their focus (e.g. by taking a post-doc abroad). Regards, Matthew –  Emerton Sep 19 '11 at 2:07
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Good point. Though I thought this was partly because in the UK you're expected to have learned the equivalent of a first-year grad curriculum in the US while you were still an undergrad. –  Noah Snyder Sep 19 '11 at 2:54
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